An Artin group is a group defined by a presentation with a finite generating set, with all relations being of the form xyxy... = yxyx... (both alternating words having the same length); at most one relation exists for each pair of distinct generators x,y. The Artin group is said to be of "large type" if each relation involves words of length at least 3.
It is conjectured that all Artin groups are biautomatic. This has been proved by Charney for Artin groups of finite type (corresponding Coxeter group finite), by Peiffer for those of extra large type (relations involve words of length at least 4), and by Brady and McCammond for Artin groups of large type with at most three generators. For general Artin groups the conjecture remains open.
In this talk, I shall briefly discuss my recent proof with Sarah Rees that all Artin groups of large type are (shortlex) automatic, and that their geodesics form a regular set. The proof is purely combinatorial. We hope to be able to address the question of their biautomaticity in the near future.
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After the seminar we will take the speaker to lunch.
See the Algebra Seminar web page for information about other seminars in the series.
James East jamese@maths.usyd.edu.au